On estimates for norms of matrix operators: the case q<p

The study of matrix operators acting between weighted sequence spaces lp,v and lq,u has become an important direction in functional analysis, particularly due to its close connection with Hardy-type inequalities and the general theory of linear operators on discrete structures. A key problem in this framework is determining when such operators are bounded and obtaining precise value or sharp estimates for their operator norms. Although considerable attention has been devoted to matrix operators whose entries satisfy the so-called Oinarov conditions, including several extensions to broader classes of kernels, the literature still lacks comprehensive norm estimates, especially in the case 1 < q < p < ∞. In this paper, we establish necessary and sufficient criteria for the boundedness of matrix operators with entries satisfying the Oinarov conditions. Furthermore, we provide both lower and upper estimates for their norms. These results not only refine previously known inequalities but also provide new tools for analyzing the structure and behavior of weighted sequence spaces. Applications of our findings include spectral characterization of matrix operators, investigation of oscillatory and non-oscillatory properties of solutions to higher-order difference equations, and the evaluation of sequences via their discrete derivatives.

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